Smoothing Properties of Bilinear Operators and Leibniz-Type Rules in Lebesgue and Mixed Lebesgue Spaces

TL;DR

This paper investigates how bilinear operators, including fractional integrals and multipliers, enhance or preserve regularity in functions within Lebesgue and mixed Lebesgue spaces, establishing smoothing properties and Leibniz-type rules.

Contribution

It provides new results on the smoothing effects of bilinear fractional integrals and multipliers, and derives Leibniz-type rules in Lebesgue and mixed Lebesgue spaces.

Findings

Bilinear fractional integral operators are smoothing, improving function regularity.

Bilinear singular multipliers preserve regularity.

Leibniz-type rules are established in Lebesgue and mixed Lebesgue spaces.

Abstract

We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.